Find the maximum and minimum values of
`(i) cos 2x + cos^(2)x`
`(ii) cos^(2) ((pi)/(4)+x)(sin x- cos x)^(2)`

Answer

To find the maximum and minimum values of the given expressions, we will break down each part step by step.

### Part (i): Find the maximum and minimum values of \( \cos 2x + \cos^2 x \)

1. **Use the identity for \( \cos 2x \)**:
\[ \cos 2x = 2\cos^2 x - 1 \]
Substitute this into the expression:
\[ \cos 2x + \cos^2 x = (2\cos^2 x - 1) + \cos^2 x = 3\cos^2 x - 1 \]

2. **Determine the range of \( \cos^2 x \)**:
The value of \( \cos^2 x \) ranges from 0 to 1. Therefore:
\[ 0 \leq \cos^2 x \leq 1 \]

3. **Substitute the extreme values of \( \cos^2 x \)**:
- When \( \cos^2 x = 0 \):
\[ 3(0) - 1 = -1 \]
- When \( \cos^2 x = 1 \):
\[ 3(1) - 1 = 2 \]

4. **Conclusion**:
The minimum value is \(-1\) and the maximum value is \(2\).

### Part (ii): Find the maximum and minimum values of \( \cos^2\left(\frac{\pi}{4} + x\right)(\sin x - \cos x)^2 \)

1. **Use the identity for \( \cos^2\left(\frac{\pi}{4} + x\right) \)**:
\[ \cos^2\left(\frac{\pi}{4} + x\right) = \frac{1 + \cos(2x)}{2} \]

2. **Express \( (\sin x - \cos x)^2 \)**:
\[ (\sin x - \cos x)^2 = \sin^2 x - 2\sin x \cos x + \cos^2 x = 1 - \sin(2x) \]

3. **Combine the expressions**:
\[ \cos^2\left(\frac{\pi}{4} + x\right)(\sin x - \cos x)^2 = \frac{1 + \cos(2x)}{2}(1 - \sin(2x)) \]

4. **Simplify the expression**:
Distributing gives:
\[ \frac{1 - \sin(2x) + \cos(2x) - \sin(2x)\cos(2x)}{2} \]

5. **Find the maximum and minimum values**:
- The maximum value of \( \sin(2x) \) and \( \cos(2x) \) is \( 1 \) and the minimum is \( -1 \).
- The expression can be evaluated at these extreme points to find the maximum and minimum values.

6. **Evaluate at extreme points**:
- When \( \sin(2x) = 1 \) and \( \cos(2x) = 1 \):
\[ \frac{1 - 1 + 1 - 1}{2} = 0 \]
- When \( \sin(2x) = -1 \) and \( \cos(2x) = -1 \):
\[ \frac{1 + 1 + 1 + 1}{2} = 2 \]

7. **Conclusion**:
The minimum value is \(0\) and the maximum value is \(2\).

### Final Results:
- For part (i): Minimum value = \(-1\), Maximum value = \(2\).
- For part (ii): Minimum value = \(0\), Maximum value = \(2\).